Friday, January 18, 2013

Colouring problem #24 - 3-colour space

Problem: Every space point is colored with exactly one of the colors red, green or blue. The sets R, G, B consist of the lengths of those segments in space with both end points red, green and blue, respectively. Show that at least one of these sets contains all non-negative real numbers.

Solution: (With reference to the diagram) Suppose the 3 colours are C1, C2 and C3. Assume that the minimum distances by which no two points with the same colour are separated (i.e. no line segment with end points of the same colour exist) be x, y and z for C1, C2 and C3 respectively. Suppose x >= y >= z. Then as shown in the diagram, any point coloured C1 would form the centre of a sphere of radius x such that the sphere can have only points coloured C2 or C3. As indicated in the diagram, a circle of diameter x√3 would then exist on the surface of the sphere such that all its points are coloured C3 and the chords of the circle form line segments of all possible lengths upto x√3, contradicting the assumption that no two points coloured C3 exist at a distance of z from each other.

Diagram for problem #24: The green circle is formed by points on surface of the sphere that are at a distance of x from the top point (on spherical surface) coloured C2, where x is the radius of the sphere centered at a point coloured C1. The diameter of the circle is xv3. If neither of the colours C1 and C2 have 2 points at a distance of x, then the circle must be coloured C3 entirely.